Construction and Linearity of Z_pZ_p^2-Linear Generalized Hadamard Codes
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The _p_p^2-additive codes are subgroups of _p^α_1×_p^2^α_2, and can be seen as linear codes over _p when α_2=0, _p^2-additive codes when α_1=0, or _2_4-additive codes when p=2. A _p_p^2-linear generalized Hadamard (GH) code is a GH code over _p which is the Gray map image of a _p_p^2-additive code. In this paper, we generalize some known results for _p_p^2-linear GH codes with p=2 to any p≥ 3 prime when α_1 ≠ 0. First, we give a recursive construction of _p_p^2-additive GH codes of type (α_1,α_2;t_1,t_2) with t_1,t_2≥ 1. Then, we show for which types the corresponding _p_p^2-linear GH codes are non-linear over _p. Finally, according to some computational results, we see that, unlike _4-linear GH codes, when p≥ 3 prime, the _p^2-linear GH codes are not included in the family of _p_p^2-linear GH codes with α_1≠0.
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